Constructing Polylogarithms on Higher-Genus Riemann Surfaces

Explore the generalization of the Brown-Levin construction of elliptic polylogarithms to Riemann surfaces of arbitrary genus in this advanced mathematics lecture. Delve into the generation of homotopy-invariant iterated integrals on higher-genus surfaces using a flat connection with simple poles in two marked points. Examine the integration kernels of the flat connection, composed of modular tensors built from convolutions of the Arakelov Green function and its derivatives with holomorphic Abelian differentials. Learn how these convolutions reproduce the Kronecker-Eisenstein kernels of elliptic polylogarithms and modular graph forms at genus one. Gain insights into ongoing research on the relationships between higher-genus polylogarithms of Enriquez-Zerbini and tensorial generalizations of Fay identities among the presented integration kernels.